User:BudjarnLambeth/Sandbox2: Difference between revisions
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{{Harmonics in cet|85.842|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning}} | {{Harmonics in cet|85.842|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning}} | ||
{{Harmonics in cet|85.842|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning (continued)}} | {{Harmonics in cet|85.842|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 14et, 11-limit WE tuning (continued)}} | ||
; [[36ed6]] | |||
* Step size: 86.165{{c}}, octave size: 1206.3{{c}} | |||
Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 36ed6 does this. | |||
{{Harmonics in equal|36|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 36ed6}} | |||
{{Harmonics in equal|36|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 36ed6 (continued)}} | |||
; [[zpi|42zpi]] | ; [[zpi|42zpi]] | ||
| Line 28: | Line 34: | ||
{{Harmonics in cet|86.329|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42zpi}} | {{Harmonics in cet|86.329|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 42zpi}} | ||
{{Harmonics in cet|86.329|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42zpi (continued)}} | {{Harmonics in cet|86.329|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 42zpi (continued)}} | ||
; [[22edt]] | ; [[22edt]] | ||
* Step size: | * Step size: 86.453{{c}}, octave size: 1210.3{{c}} | ||
Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 22edt does this. | Stretching the octave of 14edo by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning 22edt does this. | ||
{{Harmonics in equal|22|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | {{Harmonics in equal|22|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | ||
Revision as of 00:21, 28 August 2025
Title1
Octave stretch or compression
14edo benefits from octave stretch as harmonics 3, 7, and 11 are all tuned flat. 22edt, 36ed6 and 42zpi are among the possible choices.
What follows is a comparison of stretched- and compressed-octave 14edo tunings.
- 14edo
- Step size: 85.714 ¢, octave size: 1200.0 ¢
Pure-octaves 14edo approximates all harmonics up to 16 within NNN ¢.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.0 | -16.2 | +0.0 | +42.3 | -16.2 | -26.0 | +0.0 | -32.5 | +42.3 | -37.0 | -16.2 |
| Relative (%) | +0.0 | -18.9 | +0.0 | +49.3 | -18.9 | -30.3 | +0.0 | -37.9 | +49.3 | -43.2 | -18.9 | |
| Steps (reduced) |
14 (0) |
22 (8) |
28 (0) |
33 (5) |
36 (8) |
39 (11) |
42 (0) |
44 (2) |
47 (5) |
48 (6) |
50 (8) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +16.6 | -26.0 | +26.0 | +0.0 | -19.2 | -32.5 | -40.4 | +42.3 | -42.2 | -37.0 | -28.3 | -16.2 |
| Relative (%) | +19.4 | -30.3 | +30.4 | +0.0 | -22.4 | -37.9 | -47.1 | +49.3 | -49.2 | -43.2 | -33.0 | -18.9 | |
| Steps (reduced) |
52 (10) |
53 (11) |
55 (13) |
56 (0) |
57 (1) |
58 (2) |
59 (3) |
61 (5) |
61 (5) |
62 (6) |
63 (7) |
64 (8) | |
- Step size: 85.759 ¢, octave size: NNN ¢
Stretching the octave of 14edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 13-limit WE tuning and 13-limit TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +0.6 | -15.3 | +1.3 | -42.0 | -14.6 | -24.2 | +1.9 | -30.5 | -41.4 | -34.9 | -14.0 |
| Relative (%) | +0.7 | -17.8 | +1.5 | -49.0 | -17.1 | -28.2 | +2.2 | -35.6 | -48.3 | -40.7 | -16.3 | |
| Step | 14 | 22 | 28 | 32 | 36 | 39 | 42 | 44 | 46 | 48 | 50 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +18.9 | -23.6 | +28.5 | +2.5 | -16.7 | -29.9 | -37.7 | -40.8 | -39.5 | -34.3 | -25.5 | -13.4 |
| Relative (%) | +22.1 | -27.5 | +33.2 | +2.9 | -19.5 | -34.9 | -44.0 | -47.5 | -46.0 | -39.9 | -29.7 | -15.6 | |
| Step | 52 | 53 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | |
- Step size: 85.842 ¢, octave size: NNN ¢
Stretching the octave of 14edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. Its 11-limit WE tuning and 11-limit TE tuning both do this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +1.8 | -13.4 | +3.6 | -39.4 | -11.6 | -21.0 | +5.4 | -26.9 | -37.6 | -30.9 | -9.9 |
| Relative (%) | +2.1 | -15.6 | +4.2 | -45.9 | -13.6 | -24.4 | +6.2 | -31.3 | -43.8 | -36.0 | -11.5 | |
| Step | 14 | 22 | 28 | 32 | 36 | 39 | 42 | 44 | 46 | 48 | 50 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +23.3 | -19.2 | +33.0 | +7.2 | -12.0 | -25.1 | -32.8 | -35.8 | -34.4 | -29.1 | -20.2 | -8.1 |
| Relative (%) | +27.1 | -22.4 | +38.5 | +8.3 | -13.9 | -29.2 | -38.3 | -41.7 | -40.1 | -33.9 | -23.6 | -9.4 | |
| Step | 52 | 53 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | |
- Step size: 86.165 ¢, octave size: 1206.3 ¢
Stretching the octave of 14edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 36ed6 does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +6.3 | -6.3 | +12.6 | -29.0 | +0.0 | -8.4 | +18.9 | -12.6 | -22.7 | -15.4 | +6.3 |
| Relative (%) | +7.3 | -7.3 | +14.7 | -33.7 | +0.0 | -9.7 | +22.0 | -14.7 | -26.3 | -17.8 | +7.3 | |
| Steps (reduced) |
14 (14) |
22 (22) |
28 (28) |
32 (32) |
36 (0) |
39 (3) |
42 (6) |
44 (8) |
46 (10) |
48 (12) |
50 (14) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +40.1 | -2.1 | -35.3 | +25.3 | +6.5 | -6.3 | -13.8 | -16.4 | -14.7 | -9.1 | +0.1 | +12.6 |
| Relative (%) | +46.5 | -2.4 | -41.0 | +29.3 | +7.5 | -7.3 | -16.0 | -19.0 | -17.0 | -10.5 | +0.2 | +14.7 | |
| Steps (reduced) |
52 (16) |
53 (17) |
54 (18) |
56 (20) |
57 (21) |
58 (22) |
59 (23) |
60 (24) |
61 (25) |
62 (26) |
63 (27) |
64 (28) | |
- Step size: 86.329 ¢, octave size: NNN ¢
Stretching the octave of 14edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 42zpi does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +8.6 | -2.7 | +17.2 | -23.8 | +5.9 | -2.0 | +25.8 | -5.4 | -15.2 | -7.5 | +14.5 |
| Relative (%) | +10.0 | -3.1 | +19.9 | -27.6 | +6.8 | -2.3 | +29.9 | -6.3 | -17.6 | -8.7 | +16.8 | |
| Step | 14 | 22 | 28 | 32 | 36 | 39 | 42 | 44 | 46 | 48 | 50 | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -37.7 | +6.6 | -26.5 | +34.4 | +15.8 | +3.2 | -4.1 | -6.6 | -4.7 | +1.1 | +10.5 | +23.1 |
| Relative (%) | -43.7 | +7.7 | -30.7 | +39.9 | +18.3 | +3.7 | -4.8 | -7.6 | -5.5 | +1.3 | +12.1 | +26.8 | |
| Step | 51 | 53 | 54 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | |
- Step size: 86.453 ¢, octave size: 1210.3 ¢
Stretching the octave of 14edo by around NNN ¢ results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN ¢. The tuning 22edt does this.
| Harmonic | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | +10.3 | +0.0 | +20.7 | -19.8 | +10.3 | +2.8 | +31.0 | +0.0 | -9.5 | -1.6 | +20.7 |
| Relative (%) | +12.0 | +0.0 | +23.9 | -22.9 | +12.0 | +3.3 | +35.9 | +0.0 | -11.0 | -1.8 | +23.9 | |
| Steps (reduced) |
14 (14) |
22 (0) |
28 (6) |
32 (10) |
36 (14) |
39 (17) |
42 (20) |
44 (0) |
46 (2) |
48 (4) |
50 (6) | |
| Harmonic | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -31.5 | +13.2 | -19.8 | +41.3 | +22.8 | +10.3 | +3.2 | +0.8 | +2.8 | +8.7 | +18.2 | +31.0 |
| Relative (%) | -36.4 | +15.2 | -22.9 | +47.8 | +26.4 | +12.0 | +3.7 | +1.0 | +3.3 | +10.1 | +21.1 | +35.9 | |
| Steps (reduced) |
51 (7) |
53 (9) |
54 (10) |
56 (12) |
57 (13) |
58 (14) |
59 (15) |
60 (16) |
61 (17) |
62 (18) |
63 (19) |
64 (20) | |
Title2
Possible tunings to be used on each page
You can remove some of these or add more that aren't listed here; this section is pretty much just brainstorming.
(Used https://x31eq.com/temper-pyscript/net.html, used WE instead of TE cause it kept defaulting to WE and I kept not remembering to switch it)
- High-priority
13edo
- Main: "13edo and optimal octave stretching"
- 2.5.11.13 WE (92.483c)
- 2.5.7.13 WE (92.804c)
- 2.3 WE (91.405c) (good for opposite 7 mapping)
- 38zpi (92.531c)
14edo
- 22edt
- 36ed6
- 11-limit WE (85.842c)
- 13-limit WE (85.759c)
- 42zpi (86.329c)
16edo
- 25edt
- 41ed6
- 57ed12
- 2.5.7.13 WE (75.105c)
- 13-limit WE (75.315c)
- 15zpi (75.262c)
99edo
- 157edt
- 256ed6
- 7-limit WE (12.117c)
- 13-limit WE (12.123c)
- 567zpi (12.138c)
- 568zpi (12.115c)
23edo (narrow down edonoi & ZPIs)
- Main: "23edo and octave stretching"
- 36edt
- 59ed6
- 60ed6
- 68ed8
- 11ed7/5
- 1ed33/32
- 2.3.5.13 WE (52.447c)
- 2.7.11 WE (51.962c)
- 13-limit WE (52.237c)
- 83zpi (53.105c)
- 84zpi (52.615c)
- 85zpi (52.114c)
- 86zpi (51.653c)
- 87zpi (51.201c)
60edo (narrow down edonoi & ZPIs)
- 95edt
- 139ed5
- 155ed6
- 208ed11
- 255ed19
- 272ed23 (great for catnip temperament)
- 13-limit WE (20.013c)
- 299zpi (20.128c)
- 300zpi (20.093c)
- 301zpi (20.027c)
- 302zpi (19.962c)
- 303zpi (19.913c)
- 304zpi (19.869c)
- Medium priority
32edo (narrow down ZPIs)
- 90ed7
- 51edt
- 75ed5
- 1ed46/45
- 11-limit WE (37.453c)
- 13-limit WE (37.481c)
- 131zpi (37.862c)
- 132zpi (37.662c)
- 133zpi (37.418c)
- 134zpi (37.176c)
33edo (narrow down edonoi)
- 76ed5
- 92ed7
- 52edt
- 1ed47/46
- 114ed11
- 122ed13
- 93ed7
- 23edPhi
- 77ed5
- 123ed13
- 115ed11
- 11-limit WE (36.349c)
- 13-limit WE (36.357c)
- 137zpi (36.628c)
- 138zpi (36.394c)
- 139zpi (36.179c)
39edo
- 62edt
- 101ed6
- 18ed11/8
- 2.3.5.11 WE (30.703c)
- 2.3.7.11.13 WE (30.787c)
- 13-limit WE (30.757c)
- 171zpi (30.973c)
- 172zpi (30.836c)
- 173zpi (30.672c)
42edo
- 42ed257/128 (replace w something similar but simpler)
- AS123/121 (1ed123/121)
- 11ed6/5
- 34ed7/4
- 7-limit WE (28.484c)
- 13-limit WE (28.534c)
- 189zpi (28.689c)
- 190zpi (28.572c)
- 191zpi (28.444c)
45edo
- 126ed7
- 13ed11/9
- 7-limit WE (26.745c)
- 13-limit WE (26.695c)
- 207zpi (26.762)
- 208zpi (26.646)
- 209zpi (26.550)
54edo
- 86edt
- 126ed5
- 152ed7
- 38ed5/3
- 40ed5/3
- 2.3.7.11.13 WE (22.180c)
- 13-limit WE (22.198c)
- 262zpi (22.313c)
- 263zpi (22.243c)
- 264zpi (22.175c)
59edo (narrow down ZPIs)
- 93edt
- 166ed7
- 203ed11
- 7-limit WE (20.301c)
- 11-limit WE (20.310c)
- 13-limit WE (20.320c)
- 293zpi (20.454c)
- 294zpi (20.399c)
- 295zpi (20.342c)
- 296zpi (20.282c)
- 297zpi (20.229c)
64edo (narrow down ZPIs)
- 149ed5
- 180ed7
- 222ed11
- 47ed5/3
- 11-limit WE (18.755c)
- 13-limit WE (18.752c)
- 325zpi (18.868c)
- 326zpi (18.816c)
- 327zpi (18.767c)
- 328zpi (18.721c)
- 329zpi (18.672c)
- 330zpi (18.630c)
103edo (narrow down edonoi, choose ZPIS)
- 163edt
- 239ed5
- 289ed7
- 356ed11
- 381ed13
- 421ed17
- 466ed23
- 13-limit WE (11.658c)
- Best nearby ZPI(s)
118edo (choose ZPIS)
- 187edt
- 69edf
- 13-limit WE (10.171c)
- Best nearby ZPI(s)
152edo (choose ZPIS)
- 241edt
- 13-limit WE (7.894c)
- Best nearby ZPI(s)
- Low priority
111edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
125edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
145edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
159edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
166edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
182edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
198edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
212edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
243edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
247edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
- Optional
25edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
26edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
29edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
30edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
34edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
35edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
36edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
37edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
5edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
6edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
9edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
10edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
11edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
15edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
18edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
48edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
20edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
24edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)
28edo
- Nearby edt, ed6, ed12 and/or edf
- Nearby ed5, ed10, ed7 and/or ed11 (optional)
- 1-2 WE tunings
- Best nearby ZPI(s)